Contractivity properties of a quantum diffusion semigroup

TL;DR

This paper investigates the contractivity properties of a quantum diffusion semigroup, establishing inequalities that imply the decay of eigenvalues and purity, and the growth of entropy over time.

Contribution

It introduces a quantum generalization of the heat equation and proves new inequalities, including Nash and logarithmic Sobolev inequalities, for the associated semigroup.

Findings

Eigenvalues and purity decrease at least inverse polynomially in time

Entropy increases at least logarithmically in time

Ultracontractivity results are established

Abstract

We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.